Noise corrected pole and zero analyzer

Abstract

A pole and zero analyzer determines the poles and zeroes of a device by applying a stimulus signal to the device, detecting the stimulus and response signals and computing the auto- and cross-spectra. An estimated transfer function is least squares fit to a measured transfer function obtained from the auto- and cross-spectra and the poles and zeroes are obtained as the roots of the estimated transfer function. A weighting function may be used to emphasize certain desired regions of the frequencies of interest in the determination of the least squares fit. The orders of the poles and zeroes are modified until an optimum fit is reached.

We claim: 
     
       1. A method for determining poles and zeroes of an estimated transfer function of a device having a measured transfer function, comprising the steps of: stimulating the device with a stimulus signal and exciting a response signal;   detecting the stimulus and response signals;   computing from the stimulus and response signals the auto- and cross-spectra of the device at each of a plurality of frequencies of interest;   measuring a noise level incident on the cross-spectrum at each of the frequencies of interest;   generating the estimated transfer function;   generating a weighting function to emphasize certain desired regions of the estimated transfer function;   multiplying the estimated transfer function by the weighting function;   determining an error, at each of the frequencies of interest, between the measured transfer function and the estimated transfer function multiplied by the weighting function;   differentiating the error, at each of the frequencies of interest, with respect to the estimated transfer function;   equating the differentiated error to zero at each of the frequencies of interest;   determining coefficients of the estimated transfer function;   determining the poles and zeroes of the estimated transfer function as roots of the estimated transfer function; and   displaying the poles and zeroes of the estimated transfer function.   
     
     
       2. A method as in claim 1, wherein the step of detecting the stimulus and response signals further comprises the step of transforming the detected stimulus and response signals to the frequency domain. 
     
     
       3. A method as in claim 2, wherein the stimulus signal comprises random noise over the frequencies of interest. 
     
     
       4. A method as in claim 3, wherein matrix techniques are used to perform the steps of differentiating the error, equating the differentiated error and determining coefficients. 
     
     
       5. A method as in claim 4, wherein the step of generating a weighting function comprises: differentiating, with respect to frequency, the phase of the measured transfer function at each of the frequencies of interest;   converting the differentiated phase to absolute magnitude at each of the frequencies of interest;   deemphasizing any frequency of interest at which the measured noise on the cross-spectrum is high; and   normalizing to one the result of the previous step.   
     
     
       6. A method as in claim 5, further comprising after the step of differentiating the phase, the step of smoothing the phase derivative. 
     
     
       7. A method as in claim 6, further comprising after the step of converting the differentiated phase, the step of smoothing the absolute magnitude. 
     
     
       8. A method as in claim 7, further comprising after the step of converting to absolute magnitude and before the step of smoothing the phase derivative, the step of forming the square root of the absolute magnitude at each of the frequencies of interest. 
     
     
       9. A method as in claim 8, wherein the noise level incident on the cross-spectrum is measured as a coherence over a plurality of measurements at each of the frequencies of interest. 
     
     
       10. A method as in claim 4, wherein the step of generating a weighting function comprises: detecting peaks in the measured transfer function;   forming a surface at each of the detected peaks; and   summing the surfaces across the frequencies of interest.   
     
     
       11. A method as in claim 10, further comprising after the step of summing the surfaces the step of normalizing the summation of surfaces to a maximum surface value. 
     
     
       12. A method as in claim 11, wherein each surface comprises a parabola. 
     
     
       13. A method as in claim 12, wherein each parabola is centered at a detected peak in the measured transfer function. 
     
     
       14. A method as in claim 13, wherein each of the parabolas contains a substantially equal area. 
     
     
       15. A method as in claim 14, wherein each particular parabola has a width which is proportional to a width of the detected peak associated with that particular parabola. 
     
     
       16. A method as in claim 15, wherein each parabola has a height which is inversely proportional to its width. 
     
     
       17. A method as in claim 16, further comprising: determining a mean squared value of the measured transfer function;   computing an absolute magnitude squared of the measured transfer function at each of the frequencies of interest;   dividing the absolute magnitude squared by the mean squared value at each frequency of interest; and   combining the divided absolute magnitude squared with the normalized summation of parabolas.   
     
     
       18. A method for determining poles and zeroes of an estimated transfer function of a device, comprising: stimulating the device with a stimulus signal and exciting a response signal;   detecting the stimulus and response signals;   determining from the stimulus and response signals a measured transfer function of the device;   selecting an initial number of poles and an initial number of zeroes for the estimated transfer function;   determining the estimated transfer function from the initial numbers of poles and zeroes;   comparing the measured and estimated transfer functions and determining a fit from the comparison;   incrementing the numbers of poles and zeroes of the estimated transfer function until a best fit is obtained;   decrementing the number of zeroes until an optimal fit is obtained;   calculating the poles and zeroes as roots of the estimated transfer function; and   displaying the poles and zeroes.   
     
     
       19. A method as in claim 18, wherein the initial number of poles and zeroes is each one. 
     
     
       20. A method as in claim 19, wherein the step of comparing the estimated and measured transfer functions comprises: determining an error limit at each of the frequencies of interest;   subtracting the measured transfer function from the estimated transfer function at each of the frequencies of interest;   comparing the result of the subtraction to the error limit at each of the frequncies of interest; and   counting a number of results which exceed the error limit.   
     
     
       21. A method as in claim 20, wherein the step of comparing the estimated and measured transfer functions further comprises: recording the number of results exceeding the error limit; and   determining that a good fit is obtained if the recorded number is less than a predetermined acceptance number.   
     
     
       22. A method as in claim 21, wherein the optimal fit is obtained if a subsequent two decrementations of the number of zeroes result in fits that are not good. 
     
     
       23. A method for determining poles and zeroes of an estimated transfer function of a device, comprising: stimulating the device with a stimulus signal and exciting a response signal;   detecting the stimulus and response signals;   determining from the stimulus and response signals a measured transfer function of the device;   selecting an initial number of poles and an initial number of zeroes of the estimated transfer function;   determining the estimated transfer function having the initial number of poles and zeroes;   measuring an error to signal ratio of the estimated and measured transfer functions;   incrementing the numbers of poles and zeroes of the estimated transfer function if the error to signal ratio is greater than a noise to signal ratio of the measured transfer function;   determining the estimated transfer function with the incremented numbers of poles and zeroes;   calculating the poles and zeroes as roots of the estimated transfer function; and   displaying the poles and zeroes.   
     
     
       24. A method as in claim 23, further comprising, after the step of determining the estimated transfer function with the incremented numbers of poles and zeroes, the steps of: determining an effect of higher order coefficients of the estimated transfer function;   decrementing the number of zeroes of the estimated transfer function if the effect is below a predetermined level; and   redetermining the estimated transfer functon with the decremented number of zeroes.